Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes
In \textit{R. J. McEliece} and \textit{H. Rumsey jun.} [Euler products, cyclotomy, and coding, J. Number Theory 4, 302-311 (1972; Zbl 0235.12014)]\ and \textit{L. D. Baumert} and \textit{R. J. McEliece} [Weights of irreducible cyclic codes, Inform. Control 20, 158-175 (1972; Zbl 0239.94007)]\ the weight distribution of irreducible cyclic codes was studied and its connection with Gauss sums was pointed out. This connection actually goes back to the Hasse-Davenport curves. On the one hand the distribution of rational points of these curves determines the weight distribution of the associated codes. On the other hand the number of these points is expressed in terms of Gauss sums. In this paper the author relates the weight of a word in an irreducible cyclic code with the number of rational points on the Hasse-Davenport curve associated to that word. He also expresses the zeros of the zeta-functions of these curves in terms of Gauss sums, thus determining the number of rational points. However, it is in general difficult to compute Gauss sums. To do this the author considers binary irreducible cyclic codes by taking an odd integer \(n\geq 3\) such that 2 generates a subgroup of index 2 in \((\mathbb{Z}/ n\mathbb{Z})^*\) and he shows that the Gauss sums corresponding to words in an irreducible cyclic code of length \((2^{\varphi (n)/2} -1)/n\) are in a quadratic extension of \(\mathbb{Q}\). These Gauss sums can be evaluated up to an ambiguity which does not affect the weight distribution.
scientific article; zbMATH DE number 844627